Counting Approximately-Shortest Paths in Directed Acyclic Graphs

Given a directed acyclic graph with positive edge-weights, two vertices s and t, and a threshold-weight l, we present a fully-polynomial time approximation-scheme for the problem of counting the s-t paths of length at most l. We extend the algorithm for the case of two (or more) instances of the same problem. That is, given two graphs that have the same vertices and edges and differ only in edge-weights, and given two threshold-weights l 1 and l 2, we show how to approximately count the s-t paths that have length at most l 1 in the first graph and length not much larger than l 2 in the second graph. We believe that our algorithms should find application in counting approximate solutions of related optimization problems, where finding an (optimum) solution can be reduced to the computation of a shortest path in a purpose-built auxiliary graph.